About one generalization of Newmark's method for solving IDE of dynamic problems of viscoelastic linear theory. (Q2743064)

Dynamic problems of linear theory of viscoelasticity after applying one of the following methods: Galerkin method, Ritz method, finite elements method and so on, can be reduced to the system of integrodifferential equations (IDE) of the form NEWLINE\[NEWLINE M\ddot U(t)=A\left(U(t)-\int_0^tR(t-\tau)U(\tau)\,d\tau \right)F(t),\tag{*} NEWLINE\]NEWLINE with initial value conditions \(U(0)=U_0\), \(\dot U(0)=U_1\), where \(M\) and \(A\) are constant matrices, \(R(t)=\varepsilon e^{-\beta t} t^{\alpha-1}\), \(0<\alpha<1\), \(\beta>0\) and \(F(t)\) is a given function. Generalization of Newmark's method for solving (*) is developed. It is based on the representation \(U_{i+1}=U_i+ \Delta t\dot U_i+ \frac12\Delta t^2\ddot U_i+\theta\Delta t^2(\ddot U_{i+1}-\ddot U_i)\), where \(U_{i+1}=U(i\Delta t+\Delta t)\).